Stripe melting and a transition between weak and strong symmetry protected topological phases

摘要

For a gapped disordered many-body system with both internal and translation symmetry, one can define the corresponding weak and strong symmetry protected topological (SPT) phases. A strong SPT phase is protected by the internal symmetry G only while a weak SPT phase, fabricated by alignment of a strong SPT state in a lower dimension, requires additional discrete translation symmetry protection. In this paper, we construct a phase transition between weak and strong SPT phase in a strongly interacting boson system. The starting point of our construction is the superconducting Dirac fermions with pair density wave (PDW) order in 2d. We first demonstrate that the nodal line of the PDW contains a 1d boson SPT phase. We further show that melting the PDW stripe and condensing the nodal line provoke the transition from weak to strong SPT phase in 2d. The phase transition theory contains an O(4) nonlinear-σ model (NLσM) with topological Θ term emerging from the proliferation of domain walls bound to an SPT chain. A similar scheme also applies to weak-strong SPT transition in other dimensions and predicts possible phase transition from 2d to 3d topological order.